Often in medical follow-up, life testing and clinical experiments subjects may either leave a study or survive until its completion; in either case one gets to observe the age of the subject at leaving or at the completion of the study but not the true life time of the subject. In such a situation we say that the life time of the subject has been randomly censored on the right. Of course the subject may die during the study in which case we have uncensored observation. The data of this type along with the information as to which is censored or uncensored is called the randomly censored data. It is proposed to develop tests on the basis of randomly censored data of the hypothesis H sub zero: the life distribution of X is exponential (i.e., there is no aging due to cause or disease C(say) under consideration. X is life time due to cause C.) against the alternatives I) H sub 1: that it is new better than used in expectation, not an exponential (i.e., E(X-y) X greater than less than or equal to E(X) for all y greater than 0 with strict inequality for some y greater than 0.), H2: that it is new better than used, not an exponential (i.e., P(X greater than x plus y/X greater than x) less than or equal to P(X greater than y) for all x, y greater than or equal to 0 with struct inequality for some x, y greater than 0.), H3: that is has an increasing failure rate (i.e., accelerated aging due to the cause C.). It is also proposed to develop some practical estimators of the regression parameters in the multiple linear regression model when the observations are randomly censored.